Le Chatelier's Principle and the Prediction of the Effect of Temperature on solubilities R. Fernandez-Prini Deparlamento ~ u h i c de a Reactores. CNEA, Av. Libertador 8250, Buenos Aires CP-1429, Argentina
Le Chatelier's Prinriplr has received sustained attmtion in T H I SJOIIRNAI.: the rate of oubl~catiunhas heen close t o m e contribution per year for the last 25 years. In spite of such orolific literature, misconceptions and misleadinr or inexact ~tatcmcmt\are frequently included in some ot'thrm. Rrcrntlv . "The R u d n ~ rt 1 ) has published a paper i n , ~ ' ~J losr ' n N ~ lon llisusr uf Le Chatelier's Prinriple," which introduces. some confusion into the issue. Considering the educational character of THIS JOURNAL, it appears important to reflect upon some statements and conclusions in Bodner's article by analyzing the so-called Le Chatelier's Principle under a more general thermodynamic v s~ecificallv to the nersnective. Then it will be easier to a .o.~.l it . effect of tcmprraturc upon soluhility, in order toe\.i~lu~te the data which is claimed to be cmtrudietory with it 11).
.~ .
Is "Le Chatelier's Principle" a Principle? The answer to the above question requires a short historical digression. In 1888 Le Chatelier formulated his principle in the following words (2) "Tout systarne en iquilibre chimique iprouue, du fait de la uariation d'un seul des facteursde I'hquilibre, une transformationdam un sens tel que, si elle se produisoit seul, elle amanerait una uariation de signe contraire du facteur consid6ri." I t is obvious that, when Le Chatelier formulated the above orincinle his orimarv concern was tvoical .. chemical equilibria, lor which his statement provided a new and interesting predictive tool. Yow that the 1.awsofl'hermdvnamimiue firmly established and well understood, it appears that Le Chatelier's Principle can only be held as a "rule of thumb" in physical chemistry,' which has some practical value if properly applied to systems in equilibrium. As all such qualitative rules, it may prove very misleading when applied to rather extreme situations, and phase equilibria and solubility are extremes of a typically chemical equilibrium among gaseous ideal substances or pure phases. Actually, today Le Chatelier's Principle does not constitute aprinciple a t all because its consequences are fully contained in the realm of thermodynamics, hence it does not provide the scientist with any additional predictive tool. A similar case is that of the Law of Mass Action enunciated by Guldberg and Waage in 1867, it has at present only a very limited application and its main interest is historical. I t may be said that thermodynamics provides a more general framework within which both principles are found. Rieorouslv this statement is not true, however, since classicalihermddynamics cannot by itself demonstrate that the standard chemical ~otentialsare concentration independent (4). This point is ohside of the scope of the present discussion. In order to derive the expression giving the temperature In a more general perspective Le Chatelier's Principle may still be
of value. Prigogine (3)calls it the Moderation Theorem and distinguishes between systems in equilibrium or steady state which followthe Moderation Theorem, from those far from equilibrium which do not obey it. 550
Journal of Chemical Education
dependence of the equilibrium constant K, Bodner ( I ) started with the equation, AH" AS" InK=--+RT
R
then, arguing that
". . . the contribution to the equilibrium constant from entropy is temperature independent. Thus, differentiation of in K with respect to temperature, leads inevitably to the conclusion that the magnitude and sign of H o determine the effect of temperatureon the equilibrium constant" he writes, dlnK -=-
AH" dT RT2 The proper derivation of eqn. (2) does not require that ASo be temperature independent (a condition which anyhow would not suffice mathematically to derive eqn. (2) from eqn. (I),as a simple differentiation shows). I t is better to derive a more general expression for the effect of temperature upon equilibrium; this general expression will prove very useful in tackling solution equilibria. A system consisting of various phases in equilibrium, including solutions (i.e., multicomponent phases), may be represented by
The free energy change for such a process is, in terms of the chemical potentials pr,
or, in terms of activities,
K is related to the free energy change accompanying the process when it occurs with all the substances in their standard states.
Differentiating eqn. (5) with respect to temperature2 and remembering that (3plTlaT)p = -HIT2,
where
denotes the standard heat of reaction. Differentiating eqn. (4) under the conditions of equilibrium (AG = 0)
where IIai denotes the product of equilihrium activities appearing in eqn. (4). Equation (6) can be taken as the thermodynamic expression of Le Chatelier's Principle when temperature is the "factor" affectine eauilibrium. It is auite eeneral and reauires no asoro ASD sumptiok ;bout the temperature bependence of &l as would he expected of a thermodvnamic ex~ression. In order to assess the effect of temperature i n the solubility of salts and gases which is our main goal, it is necessary to consider some other relationships. The soluhility equilibria we shall be dealing with involve a pure solute (either a gas or a solid) in equilihrium with its saturated solution. Such equilibrium may be depicted by
Table 1. Heats of Solution of Simple Gases in Some Liquids at 298.2 K
A%, Ne
CsHe H20 CsHs CCl, H20
NMA" HL
CsHe CCl,
NMAd H,0
As the temperature changes, the equilihrium concentration of the solute%c will also change. Consequently,
From eqns. (7) and (8) one gets
The activity scale is chosen such that the reference state is infinite dilution (Henry's law reference state). AffYol= ff; Hi is the differential heat of solution at infinite dilution. Since az = f 2 ~ 2
From eqns. (9) and (10) the final expression for the temperature dependence of the soluhility becomes,
where AEsOl= ff2 - H; is the heat of solution a t saturation. The derivation of eun. (11)has reuuired no assnm~tionson the type ot'interactiu& occurring i n the solutions;qn. (11) sho~lldnpply to all known systems. Particular assumptions about va&i types of behaviors or interactions may be helpful in order to relate the variables of eqn. (11)with other thermodynamic quantities which may he more easily amenahle to experimental determination. Equation (11) possesses all the information which is contained in Le Chatelier's Principle as applied to solubility equilibria, moreover i t avoids the shortcomings of simpler 0, the saturation conformulations. Since (a In U Z / ~ C Z>) T centration c? increases (decreases) with t e m ~ e r a t u r ewhen A& is pos&ve (negative). That is, when theprocess of dissolution of one mole of solute in an infinite amount of saturated solution is endothermic (exothermic). Thus, it is evident that the classical formulation of Le Chatelier's Principle is more restrictive than the general thermodynamic formulation of the effect of temperature upon systems in equilibrium given by eqn. (11).In order to use eqn. (11) it is necessary to have a clear understanding of the composition of the phases which are in equilibrium; this point will he further elaborated below. Solublllty of Gases In Liquids: The Effect of Temperature The solubility of simple gases (HZ,N2,02, noble gases) in liquids, is correctly expressed by Henry's law = k2x2
Solvent
Ar
Pure Solute + Saturated Solution
~2
Gas
(12)
Strictly Kis pressure independent only if the standard states are the ideal gases at 1 atm. For other standard states, as usually employed for condensed phases. Kdepends to a small extent on p. c2 denotes any temperature independent concentration scale, e.g., molaiity or molar fraction.
cal mol-'
Reference
1.860 -1,910 420 -74 -2.820 100 980 626 500 -1.000
a c a B
c b
a a b
c
~iwoni.
R. A.. J ~ h pchem.. . 67,1840 (1963). 'Wood. R. H.. and Debney. 0. E.. J. Phys Chem. 12.4651 (19681. cPieroni, R. A., J. Phys Chem.. 69, 261 (1968). a
NMA: Nmethylacetarnide.
when the pressure does not exceed about 2 MPa. In eqn. (12) pz denotes the partial pressure of the gas and kz is Henry's constant. In order to evaluate the effect of temperature, pz has to be fixed and it is most convenient to do so for 1atm, which is the standard state if the gas is considered ideal. For that value of pz, the saturation mole fraction is equal to llkz. Equation (11) becomes,
for systems where Henry's law applies; this is the case for the solution of simple gasesin liquid>because x2 is very small. For AH;,l,because in eqn. (10) (d In these ideal solutions &,I= azldT),, = 0 and H; = H z . In order to predict the sign of AFfgland thence the effect of temperature on gas solubilities, Bodner (I) divided the dissolution process into two steps, AH0
Step a: Pure gaseous solute +Pure liquid solute AHb
Step b: Pure liquid solute +Saturated
solution
He argues that AHa< 0 and AH,, z 0 thus AEsol< 0 and the solubility of gases in liquids should decrease with increasing temperature. As an illustration, Table 1reports some examples of solutions of simple gases in liquids having ARs,l> 0. i.e.. accordine- to e m . . (13) . . their solubilitv increases with temperature. This apparent failure in predicting AE,l and, consequently, the temperature dependence of the soluhility of gases is due to the assumptions made regarding the process of dissolution as described above. That scheme may be applied in its more sim~listicwav onlv when the temperature is such that easeous a n d liquid solutemay be in eqilibrium (near their normal boiling point). Prausnitz t5) has employed a similar scheme hut on a mure general basis. His assumptions reaardinr- AH, - and All,, are less sweeping and they are more c&efully evaluated. ~ h u sfor , gases a t temperatures higher than the corresponding critical temperatures, there is a single fluid phase, hence Step a can only refer to an increase in the density of the gas. When the gas temperature is above its inversion temperature (the temperature a t which the Joule-Thompson coefficient becomes zero), AHH.> O;furthermore, A H b may be of comparable magnitude to AH;l. Consequently, amore careful consideration of the enthalpic changes associated with Steps a and b shows that, depending on the characteristics of the system, the heat of solution may be either positive or negative. A molecular model for the dissolution of gases in liquids also may he employed successfully to predict the sign of Volume 59 Number 7 July 1982
551
Table 2. Heats of Solution of Anhydrous and Hydrated Alkali Metal Halides and Hydroxides at 298.2 K ( 7) Electrolyte
n=0
MY
- 8.850
LiCl
Liar
-11.670 -15.130 - 5.632 - 144
Lil LiOH
NaBr
-
Nal NaOH KF KOH RbF RbOH
CsOH
n=
1
-4.560' -5,560 -7.090 -1.600e
n=3
n=2
-2,250'
-3.530
t140'
+4,454s
1.800
+3,85Sa
-10.637 - 4.238
-5.118'
-13.769 6.240 -14,900 8,810 -17.100
-3.500
-
CsF
A ~ & ( M X ~ ~ H ~ mol-' O)/~~I
- 100 -4,310 -2.500 -4.900
+1.666* -2.500 (1'/2H20)a t320(1%H20) t210
-1,300(1%H20)
Integral heat of solution against ndm fa NaBr in water (8) 0 Experimnla paints (8).- - -: Slope of the curve at saturation (5).
Denotes the stable crystalline phase at 298 K according to ref. 9.
Eley (6) has also divided the solution process into two steps, Step c:
Formation of a cavity in the host liquid (AH,).
Step d:
Introduction of an isolated gas molecule into the cavitity (AHd.
AH, is positive and AHd < 0 because it is directly related to the attractive energy between the guest particle and the molecules_of the solvent in the walls of the cavity. Finally, the sign of AH;l is established by delicate balance between the thermal energies involved in both steps. This versatile model explains the fact that the same gases which in normal liquids have a solubility that increases with temperature have in water a t room tem~eraturea solubility that decreases with temperature (AHmI< 0, cf. Table 1). This is attributed by Eley (6) to the open structure of the solvent water in the neighborhood of room temperature, which makes AH, 0. I t is important to realize that shortcomings in the predictive capacity of either model do not imply that eqn. (11) is inexact. A model is only a shortcut to avoid cumbersome mathematical treatments; it should, however, be selected with a physical feeling for the problem; but there always is some degree of uncertainty about its performance.
-
Temperature Coefficient of the Solublllty of Soluble Strong Electrolytes For these hinarv svstems. e m . (11)cannot be simplified . because saturated electrolyte solutions with solutes such as NaOH. NaCI. etc. are verv far from ideal solutions. However, the&& of d c ' ~ / dis~stillgiven by ms,l, the heat of solution of one mole of salt in a large quantity of saturated solution. Bodner (I) has rightly pointed out the difficulties-of quantitative prediction of dc&T for soluble salts. Since AH,l = Rz - H,", and AH:oIis g~erall?javailahle,the problem consists in determining Lz = Hml- H & the relative partial molal heat content of the solute at saturation. There is a further difficulty in this case, in order to proceed to a quantitative application of eqn. (11) (a In a z / d c z ) ~has to be known a t saturation. This last factor, while not changing the sign of (dcz/dT) may modify appreciably the predicted value. The discrevancies vointed out hv Bodner (1) between prrdirted and;bsened changes of sol"bility with temperature are inexact and misleading. They arise mninly due tu the use of the incorrect composition for the crystalline salt which is in thrrmodvnamir eauilibrium with the saturated snlutions a t room temperature. for alkali metal halides and Table 2 gives the values of hydroxides formicg stable hydrates at room temperature (7). The values of A H Z used by Bodner (I) correspond to the
.
m;I +
mLI
552
Journal of Chemical Education
anhydrous salts (cf. the Table in ref. ( I )) which, for all the cases illustrated in Table 2, are not the stable phase of the crystalline salt at room temperature. In order to apply any thermodynamic relation to describe equilibria, it is necessary to have a clear understanding of the nature and comoosition of the snecies involved in the eauilib&. &herwise shortcomings in our knowledge u t the &stems mav be attributrd to failure in the uredictiw r;ivaritv . . ~ ,the i therniodynomic relations. 'I'ablc 2 shows that for NaOH. Aff4, is 5,519 cal mol-I more endot hermic for the stublt, mon~hydratethan fur thr anhyd r m s salt.'l'hus, the value of -2 kcal mo-1 r i ~ e nhy Bodner for a77,,1 wouldbecome -+3 kcal mol-I if the correct solid species is considered. Hence, the observed increase in solubility for NaOH with temperature is correctly predicted by eqn. (11). This point is illustrated with one example mentioned by Bodner. The solubility of NaBr is okserved to increase with temperature. Wallace (8) measured Lz for NaBr aqueous solutions up to saturation. In order to relate the integral heat of dissolution per mole (AH.,l/nz), which is the experimental quantity, to the relative partial molal heat contents of solute and solvent, it is convenient to write,
The figure is a plot of AH.,lln~ against n h for N e r in water a t 298 K. At any point of the curve the slope gives_ll and the intercept of the slope with the ordinate axis gives L_2 From the example in Figure 1it is quite clear that L Zchanges with concentration in a complicated and unpredictable way, which makes it verv difficult to extrapolate its value a t saturation from value; for lower concentrations. For NaBr, Ez would have the largest negative value (-980 cal mol-I) a t about 6 molal, becoming -480 cal mol-I at saturation (9.19 m).For this electrolyte Lhe contribution of Lz to is exothermic and if the AHz1 for the proper hydrate (NaBr. 2HzO) is not employed, it would appear that the contribution of Ez worsens the agreement with the predictions of eqn. (ll), as claimed by Bodner. The largest contribution t o the aovarent discrevancv .. . . hetween Aii'~i',~ fnr the solid salts and theobser\.ed temperature variation ot'solubilitv is that due to thr incorrect choice of the crystalline phase in equilibrium; Ez is always relatively small, albeit not neeliaible. For the alkgline earth salts the situation is even more dramatic. For two of the cases mentioned by Bodner, the difference between the stable crystalline hexahydrate heat of solution a t infinite dilution and that of the anhydrous salts are 34 kcal mol-' (MgCl2) and 25 kcal mol-I (Mg(N03)~)more
+
m.,~
endothermic; this large difference of heat contents will affect significantly the predictions of eqn. (11). On the other hand, the tables of solubility data ( 9 ) show clearly that when the anhydrous salts become the thermodynamically stable crystalline form, the increase of solubility with temperature is remarkably smaller. In conclusion, it maybe said thateqn. (11) correctly predicts the observed changes of solubility with temperature from the heats of solution. Failures of models and incorrect choice of the components of the systems in equilibrium are usually responsible for the apparent discrepancies frequently claimed
between the predictions of eqn. (11) and the observed behavior of the systems. Literature Cited 111 nodner. T..M.. J.CHEM. EDVC.. 51,117 119801. 121 M m e . W. .I.. "Phy~icslChemistry: Prentice-Hall Inc., Englewod Cliffs. Nd. 1972. p. 291. (3) Glansdorf, P..and P ~ i g w i n e , "Thermodynamic Theow of Structure. Stability and Fluctuations," Wiley~lnterscienee.Now York. 1971, p. 6 1 14) McClasl,an, M. 119fi61. (51 prsm.ih, J. ~ . , ~ ~ ~ ~ ~ ~ ~ ~ i ~ ~ ~ h Enzleuod Clifis. NJ. 1969. p 362. ,,( Eley,D,D ,,Tra nr, dnyS,,c,,35,1281 ,1939), 171 parker. V. B . . N = ~ Istd. . R ~ ID. U ~ OSP,., 2 (1965). 181 w a k e . E..J. Amer Chem Soc., 71,2486 119491. (91 Linke. W. F.. "Sulubilities: Inorjanic and Metal Organic Compuunda." A.C.S.. W a ~ h ~ ington. 1965.
,,,
I..
N.B.s..
w.
Volume 59
Number 7
July 1982
553
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